I'm just trying to learn induction from scratch and my book asks me to proof the generalized version of the triangle inequality.
If
are arbitrary real numbers, then
My solution is as follows:
Let
be the preposition that
and
are true.
is the normal triangle inequality, so i assume i don't need to show that it is true.
With the induction hypothesis i assume that
is true.
Setting
So
We are trying to show that
Using cases
1)
and
this is true because
according to the induction hypothesis. Also according to the transitive property of real numbers if a< b, then a+c < b+c.
2)
and
Since
is true then
3
and
Therefore it is true that
Is my proof correct ?
And is there a better way of doing this?